Mathematics Education
Last modified on June 5, 2009.
Write-Up #6
Examine each of the following 11 explorations. Select any one or a combination of them for a write-up.
1. Construct a triangle and its medians.
Construct a second triangle with the three sides having the lengths
of the three medians from your first triangle. Find some relationship
between the two triangles. (E.g., are they congruent? similar?
have same area? same perimeter? ratio of areas? ratio or perimeters?)
Prove whatever you find.
2. If the original triangle
is equilateral, then the triangle of medians is equilateral. Will
an isosceles original triangle generate and isosceles triangle
of medians? Will a right triangle always generate a right triangle
of medians? What if the medians triangle is a right triangle?
Under what conditions will the original triangle and the medians
triangle both be right triangles?
3.
4. A 4 by 4 picture hangs on
a wall such that its bottom edge is 2 ft above your eye level.
How far back from the picture should you stand, directly in front
of the picture, in order to view the picture under the maximum
angle? Side view:
5. The football rules in college
football were changed a few years ago have made the uprights 5
feet narrower than previously. Many game commentators have harped
about how much harder it is to kick field goals from the hash
marks. Assume the field goal is attempted from the hash marks.
At what yard marker does the kicker have maximum angle to the
two uprights. Note: You will need to find out the width of the
uprights and the width of the hash marks . . . make a sketchpad
model. Is there any merit to some commentators argument to take
a penalty in order to have a "better angle" on the field
goal kick?
6. Given three points A, B,
and C. Draw a line intersecting AC in the point X and BC in the
point Y such that
7. Construct the common tangents
to two given circles. Make a script tool. Test it for all the
different cases. Or, do you need different script tools for the
different cases?
8. Of a triangle, given two
vertices A and B, and the angle at the third vertex C (the angle
opposite side AB). What is the locus of the point C? See Script
# 18 in Assignment 5.
(Note: Think of this as turning a fixed angle so that its sides always rest on the two endpoints of the segment AB. Remember that the size of the angle is fixed but the sides can move along A and B simultaneously.)
9. A parabola is the set of
points equidistant from a line, called the directrix, and a fixed
point, called the focus. Assume the focus is not on the line.
Construct a parabola given a fixed point for the focus and a line
(segment) for the directrix.
a. Use an Action Button to generate the parabola from an animation and trace of a constructed point.
b. Repeat 9a with a trace of the tangent line at the constructed point.
c. Use the locus command to generated the parabola from a constructed point or the tangent line at that point.
10. Construct the locus of points
equidistant from a fixed point F and a circle. In other words,
repeat the parabola construction but use a circle as the "directrix."
Let F be any point in the plane other than the center of the circle.
Assume F is not on the circle; it can be either inside or outside.
11. Consider any triangle ABC.
Find a construction for a point P such that the sum of the distances
from P to each of the three vertices is a minimum.
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